How to draw edges on a sphere to make parts that fit into each other
You work with too many polygons and you should think another way. If we try something like your last picture:

Create a UV sphere, stretch it on the Z axis, keep only 1/32 that you mirror on the X and Z axis:



Cut the hole with a knife:



Extrude inwards:



Use an Array in Object Offset mode to make the mesh rotate around an empty, work on the mesh as long as necessary before applying the Mirror and Array modifiers: